The error "Mailbox quota exceeded" is generated when a mailbox is over quota. To correct such an issue, login to cPanel and access the email accounts section.
Once inside, you can use the modify quota option to increase the size limit of the account. Once completed, mail should begin to arrive moments later. No incoming emails will be lost if you run over quota.
You should consider deleting older messages to avoid having to increase your quota frequently, or consider setting your quota far higher than necessary. Allocating additional amounts of storage to a mailbox will not affect account usage.
Other relevant error messages:
mailbox is full - retry timeout exceeded